Alphabeta MathHamiltonian Mechanics and Completely Integrable Systems
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 38 results · all verified · 38 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 38 also cleared it.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
Differential Geometry
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Symplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Symplectic vector field
Proposition
A vector field is symplectic iff ι_Xω is closed
Definition
Hamiltonian vector field and Hamiltonian function
Theorem
Hamiltonian vector fields exist uniquely for smooth functions
Proposition
Hamiltonian vector fields are symplectic and symplectic fields are locally Hamiltonian
Proposition
Hamiltonians for a fixed vector field differ by a locally constant function
Theorem
Symplectic vector fields modulo Hamiltonian vector fields are first de Rham cohomology
Theorem
Hamiltonian flows preserve the symplectic form
Proposition
A Hamiltonian is conserved along its own flow
Definition
Poisson bracket on a symplectic manifold
Proposition
The Poisson bracket is bilinear, skew, and a derivation in each entry
Theorem
The Hamiltonian vector-field map is a Lie antihomomorphism
Theorem
The Poisson bracket satisfies the Jacobi identity
Theorem
Smooth functions form a Poisson algebra
Proposition
Observable evolution equation
Definition
First integral and Poisson-commuting functions
Proposition
F is a first integral of H iff F and H Poisson commute
Theorem
Hamiltonian flows commute iff their Hamiltonians Poisson commute up to locally constant bracket
Theorem
Hamilton equations in canonical cotangent coordinates
Proposition
Coordinate formula for the Poisson bracket
Proposition
The cotangent lift of a vector field is Hamiltonian
Definition
Time-dependent Hamiltonian vector field and flow
Proposition
Time-dependent Hamiltonian evolution is symplectic
Definition
Canonical transformation
Theorem
Liouville volume preservation
Corollary
Hamiltonian flow has zero divergence with respect to symplectic volume
Definition
Liouville vector field on an exact symplectic manifold
Proposition
The canonical Liouville vector field on a cotangent bundle is radial in momenta
Corollary
Poincaré recurrence for finite-volume Hamiltonian invariant regions
Definition
Lagrangian action functional on curves
Theorem
Euler–Lagrange equations
Definition
Fibre derivative or Legendre map of a Lagrangian
Definition
Regular and hyperregular Lagrangian
Definition
Energy and Hamiltonian of a hyperregular Lagrangian
Theorem
Equivalence of Euler–Lagrange and Hamilton equations for hyperregular Lagrangians
Proposition
A natural mechanical Lagrangian gives the kinetic-plus-potential Hamiltonian
Definition
Completely integrable Hamiltonian system
Proposition
Regular common level sets are Lagrangian submanifolds
Proposition
Commuting Hamiltonian vector fields integrate to a local ℝⁿ-action
Lemma
Stabilizer of the ℝⁿ-action on a compact connected regular fibre is a full lattice
Theorem
Compact connected regular fibres are tori
Definition
Action and angle coordinates
Theorem
Liouville–Arnold action–angle theorem
Corollary
Motion of a completely integrable Hamiltonian is linear on invariant tori
Proposition
Period-lattice monodromy obstructs global action–angle coordinates
False statement
Every symplectic vector field has a global Hamiltonian function
False statement
Hamiltonian functions for one vector field differ by one global constant on a disconnected manifold
False statement
H↦ X_H is a Lie homomorphism under the library Poisson convention
False statement
Hamiltonian flows are complete on every symplectic manifold
False statement
n independent first integrals automatically form a completely integrable system
False statement